The problem asks to evaluate the limit $\lim_{x \to 3} f(x)$ based on the provided graph of the function $f(x)$.

Step-by-step analysis:

1. Behavior near $x = 3$: We observe that as $x$ approaches 3 from the left and right, the function $f(x)$ appears to approach a value, but there is a dot at $y = -3$ indicating a hole or discontinuity.

2. Left-hand and right-hand limits:

   • As $x \to 3^{-}$ (from the left), the function values seem to get closer to $y = -3$.
   • As $x \to 3^{+}$ (from the right), the function also appears to approach $y = -3$.

3. Discontinuity: The point $(3, -3)$ is shown as a solid dot, meaning the function might be discontinuous at $x = 3$. However, this does not affect the limit; we only care about the behavior as $x$ approaches 3, not the actual value of $f(3)$.

Conclusion:

Both the left-hand and right-hand limits approach $y = -3$, so the overall limit is $-3$.

Final Answer:

$\lim_{x \to 3} f(x) = -3$ The correct choice is -3.

Would you like any additional details or have any further questions?

Here are five related questions you can explore:

1. What is the significance of the solid dot at $x = 3$?
2. How do left-hand and right-hand limits affect the existence of a limit?
3. What is the difference between a limit and the function value at a point?
4. What types of discontinuities can occur, and how do they impact limits?
5. How do you handle limits at infinity based on graphical behavior?

Tip: When evaluating limits graphically, always check both sides (left and right) to confirm whether they approach the same value.